Elementary linear algebra

BIOLOGY AND LIFE SCIENCES

Age distribution vector, 372, 383, 384, 387

Age progression software, 174

Age transition matrix, 372, 383, 384, 387

Agriculture, 37, 50, 101

Cosmetic surgery results simulation, 174

Duchenne muscular dystrophy, 359

Galloping speeds of animals, 270

Genetics, 359

Health care expenditures, 140

Heart rhythm analysis, 249

Hemophilia A, 359

Hereditary baldness, 359

Nutrition, 11

Population

of deer, 37

of laboratory mice, 95

of rabbits, 373

of sharks, 388

of small fish, 388

Population genetics, 359

Population growth, 372, 373, 383, 384,

387, 388, 390

Predator-prey relationship, 388

Red-green color blindness, 359

Reproduction rates of deer, 97

Sex-linked inheritance, 359

Spread of a virus, 95

Vitamin C content, 11

Wound healing simulation, 174

-linked inheritance, 359

BUSINESS AND ECONOMICS

Borrowing money, 23

Computer operator charges, 139

Demand

for a grade of gasoline, 97

for a rechargeable power drill, 97

Demand matrix, external, 91

Economic system, 90, 91

of a small community, 96

Finance, 23

Gasoline sales, 99

Industrial system, 96, 101

Input-output matrix, 90

Leontief input-output model(s), 90, 91, 97

Major League Baseball salaries, 101

Manufacturing

labor and material costs, 99

models and prices, 144

production levels, 51, 99

Net profit, Microsoft, 32

Output matrix, 91

Profit, from crops, 50

Purchase of a product, 95

Revenue

fast-food stand, 236

General Dynamics Corporation, 260, 270

Google, Inc., 286

telecommunications company, 236

textbook publishers, 137

Sales, 37

concession area, 42

Wal-Mart, 32

Satellite television service, 85, 86, 141

Textbook publishing, 137

ENGINEERING AND TECHNOLOGY

Aircraft design, 79

Circuit design, 316

Computer graphics, 332

Computer monitors, 184

Control system, 308

Controllability matrix, 308

Cryptography, 87–89, 95–97, 100, 101

Data encryption, 87

Decoding a message, 89, 96, 97, 100, 101

Digital signal processing, 166

Electrical network analysis, 30, 31, 34, 37,

144

Electronic equipment, 184

Encoding a message, 88, 95, 97, 100

Encryption key, 87

Engineering and control, 124

Error checking

digit, 194

matrix, 194

Feed horn, 217

Global Positioning System, 16

Image morphing and warping, 174

Information retrieval, 58

Internet search engine, 58

Ladder network, 316

Locating lost vessels at sea, 16

Movie special effects, 174

Network analysis, 29–31, 33, 34, 37

Radar, 166

Sampling, 166

Satellite dish, 217

Smart phones, 184

Televisions, 184

Wireless communications, 166

MATHEMATICS AND GEOMETRY

Adjoint of a matrix, 128, 129, 136, 140, 144

Collinear points in the -plane, 133, 137

Conic section(s), 220, 223

general equation, 135

rotation of axes, 215–218, 220, 223,

377–379, 384, 387

Contraction in 331, 335, 336

Coplanar points in space, 134, 137

Cramer’s Rule, 124, 130, 131, 136, 137, 140

Cross product of two vectors, 271–274,

282, 283, 286

Differential equation(s)

linear, 212, 219, 220, 223

second order, 158

system of first order, 348, 374, 375,

383, 384, 387, 388, 390

Expansion in 331, 335, 336, 339

Fibonacci sequence, 388

Fourier approximation(s), 279–281, 283, 286

Geometry of linear transformations in

330–332, 335, 336, 339

Hessian matrix, 369

Jacobian, 108, 139

Lagrange multiplier, 34

Laplace transform, 124

Least squares approximation(s), 275–278, 283

linear, 276, 283, 286

quadratic, 277, 283, 286

Linear programming, 47

Magnification in 335, 336

Mathematical modeling, 267, 268, 270

Parabola passing through three points, 144

Partial fraction decomposition, 34, 37

Polynomial curve fitting, 25–28, 32, 34, 37

Quadratic form(s), 376–382, 384, 387, 390

Quadric surface, rotation of, 382, 384

Reflection in 330, 335, 336, 339, 340

Relative maxima and minima, 369

Rotation

in 297, 337, 385, 389

in 333, 334, 336, 339

Second Partials Test for relative extrema, 369

Shear in 331, 332, 335, 336, 339

Taylor polynomial of degree 1, 276

Three-point form of the equation of a

plane, 135, 137, 140

Translation in 302, 337

Triple scalar product, 282

Two-point form of the equation of a line,

133, 137, 140, 144

Unit circle, 247

Wronskian, 213, 219, 220, 223

R2,

R2,

R3,

R2,

R2

,

R2,

R2

,

R2

,

R2

,

xy

X

INDEX OF APPLICATIONS

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PHYSICAL SCIENCES

Acoustical noise levels, 28

Airplane speed, 11

Area

of a parallelogram using cross product,

273, 274, 282, 286

of a triangle

using cross product, 283

using determinants, 132, 137, 140,

144

Astronomy, 27, 268

Balancing a chemical equation, 4

Beam deflection, 64, 72

Chemical

mixture, 37

reaction, 4

Computational fluid dynamics, 79

Crystallography, 207

Degree of freedom, 158

Diffusion, 348

Dynamical systems, 388

Earthquake monitoring, 16

Electric and magnetic flux, 234

Flexibility matrix, 64, 72

Force

matrix, 72

to pull an object up a ramp, 151

Geophysics, 166

Grayscale, 184

Hooke’s Law, 64

Kepler’s First Law of Planetary Motion,

135

Kirchhoff’s Laws, 30, 316

Lattice of a crystal, 207

Mass-spring system, 158, 161

Mean distance from the sun, 27, 268

Natural frequency, 158

Newton’s Second Law of Motion, 158

Ohm’s Law, 316

Pendulum, 219

Planetary

orbits, 135

periods, 27, 268

Primary additive colors, 184

RGB color model, 184

Stiffness matrix, 64, 72

Temperature, 34

Torque, 271

Traffic flow, 28, 33

Undamped system, 158

Unit cell, 207

end-centered monoclinic, 207

Vertical motion, 37

Volume

of a parallelepiped, 282, 283, 286

of a solid, 108

of a tetrahedron, 134, 137

Water flow, 33

Weather, 325

Work, 242

SOCIAL SCIENCES AND

DEMOGRAPHICS

Cellular phone subscribers, 101

Consumer preference model, 85, 86, 95,

141

Doctoral degrees awarded, 270

Energy consumption, 285

Final grades, 100

Motor vehicle registration, 97

Politics, voting apportionment, 51

Population

of consumers, 95

regions of the United States, 51

of smokers and nonsmokers, 95

United States, 32

world, 267

Population migration, 100

Smokers and nonsmokers, 95

Sports, Super Bowl I, 36

Television watching, 95

Test scores, 102

STATISTICS AND PROBABILITY

Canonical regression analysis, 298

Least squares regression

analysis, 92–94, 97, 101, 259, 265–270

cubic polynomial, 270

line, 93, 97, 101, 265, 268, 270, 290

quadratic polynomial, 267, 270

Markov chain, 325

Multiple regression analysis, 298

Multivariate statistics, 298

State matrix, 85, 100, 141, 325

Steady state probability vector, 386

Stochastic matrices, 84–86, 95, 97, 100, 325

MISCELLANEOUS

Architecture, 382

Catedral Metropolitana Nossa Senhora

Aparecida, 382

Determining directions, 16

Dominoes, A2

Flight crew scheduling, 47

Sudoku, 114

Tips, 23

U.S. Postal Service, 194

ZIP + 4 barcode, 194

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Elementary Linear Algebra

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Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Elementary Linear Algebra

Ron Larson

The Pennsylvania State University

The Behrend College

Seventh Edition

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Elementary Linear Algebra

Seventh Edition

Ron Larson

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Systems of Linear Equations 1

1.1 Introduction to Systems of Linear Equations 2

1.2 Gaussian Elimination and Gauss-Jordan Elimination 13

1.3 Applications of Systems of Linear Equations 25

Review Exercises 35

Project 1 Graphing Linear Equations 38

Project 2 Underdetermined and Overdetermined Systems 38

Matrices 39

2.1 Operations with Matrices 40

2.2 Properties of Matrix Operations 52

2.3 The Inverse of a Matrix 62

2.4 Elementary Matrices 74

2.5 Applications of Matrix Operations 84

Review Exercises 98

Project 1 Exploring Matrix Multiplication 102

Project 2 Nilpotent Matrices 102

Determinants 103

3.1 The Determinant of a Matrix 104

3.2 Determinants and Elementary Operations 112

3.3 Properties of Determinants 120

3.4 Applications of Determinants 128

Review Exercises 138

Project 1 Stochastic Matrices 141

Project 2 The Cayley-Hamilton Theorem 141

Cumulative Test for Chapters 1–3 143

Vector Spaces 145

4.1 Vectors in 146

4.2 Vector Spaces 155

4.3 Subspaces of Vector Spaces 162

4.4 Spanning Sets and Linear Independence 169

4.5 Basis and Dimension 180

4.6 Rank of a Matrix and Systems of Linear Equations 189

4.7 Coordinates and Change of Basis 202

4.8 Applications of Vector Spaces 212

Review Exercises 221

Project 1 Solutions of Linear Systems 224

Project 2 Direct Sum 224

Rn

v

1

2

3

4

Contents

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Inner Product Spaces 225

5.1 Length and Dot Product in 226

5.2 Inner Product Spaces 237

5.3 Orthonormal Bases: Gram-Schmidt Process 248

5.4 Mathematical Models and Least Squares Analysis 259

5.5 Applications of Inner Product Spaces 271

Review Exercises 284

Project 1 The QR-Factorization 287

Project 2 Orthogonal Matrices and Change of Basis 288

Cumulative Test for Chapters 4 and 5 289

Linear Transformations 291

6.1 Introduction to Linear Transformations 292

6.2 The Kernel and Range of a Linear Transformation 303

6.3 Matrices for Linear Transformations 314

6.4 Transition Matrices and Similarity 324

6.5 Applications of Linear Transformations 330

Review Exercises 337

Project 1 Reflections in (I) 340

Project 2 Reflections in (II) 340

Eigenvalues and Eigenvectors 341

7.1 Eigenvalues and Eigenvectors 342

7.2 Diagonalization 353

7.3 Symmetric Matrices and Orthogonal Diagonalization 362

7.4 Applications of Eigenvalues and Eigenvectors 372

Review Exercises 385

Project 1 Population Growth and Dynamical Systems (I) 388

Project 2 The Fibonacci Sequence 388

Cumulative Test for Chapters 6 and 7 389

Complex Vector Spaces (online)*

8.1 Complex Numbers

8.2 Conjugates and Division of Complex Numbers

8.3 Polar Form and DeMoivre’s Theorem

8.4 Complex Vector Spaces and Inner Products

8.5 Unitary and Hermitian Matrices

Review Exercises

Project Population Growth and Dynamical Systems (II)

R2

R2

Rn

vi Contents

5

6

7

8

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Linear Programming (online)*

9.1 Systems of Linear Inequalities

9.2 Linear Programming Involving Two Variables

9.3 The Simplex Method: Maximization

9.4 The Simplex Method: Minimization

9.5 The Simplex Method: Mixed Constraints

Review Exercises

Project Cholesterol Levels

Numerical Methods (online)*

10.1 Gaussian Elimination with Partial Pivoting

10.2 Iterative Methods for Solving Linear Systems

10.3 Power Method for Approximating Eigenvalues

10.4 Applications of Numerical Methods

Review Exercises

Project Population Growth

Appendix A1

Mathematical Induction and Other Forms of Proofs

Online Technology Guide (online)*

Answer Key A7

Index A39

*Available online at www.cengagebrain.com.

Contents vii

9

10

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Welcome to the Seventh Edition of Elementary Linear Algebra.

My primary goal is to present the major concepts of linear

algebra clearly and concisely. To this end, I have carefully

selected the examples and exercises to balance theory with

applications and geometrical intuition. The order and coverage

of topics were chosen for maximum efficiency, effectiveness,

and balance. The new design complements the multitude of

features and applications found throughout the book.

New To This Edition

NEW Chapter Openers

Each Chapter Opener highlights five real-life applications

of linear algebra found throughout the chapter. Many of the

applications reference the new Linear Algebra Applied featured

(discussed below). You can find a full listing of the applications

in the Index of Applications on the inside front cover.

NEW Linear Algebra Applied

Linear Algebra Applied describes a real-life application

of concepts discussed in a section. These applications include

biology and life sciences, business and economics,

engineering and technology, physical sciences, and statistics

and probability.

NEW Capstone Exercises

The Capstone is a conceptual problem that synthesizes

key topics to check students’ understanding of the section

concepts. I recommend it.

ix

Preface

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REVISED Exercise Sets

The exercise sets have been carefully and extensively

examined to ensure they are rigorous, relevant, and

cover all topics suggested by our users. The exercises

have been reorganized and titled so you can better

see the connections between examples and exercises.

Many new skill building, challenging, and application

exercises have been added. As in earlier editions, the

following pedagogically-proven types of exercises

are included:

• True or False Exercises ask students to give examples

or justifications to support their conclusions.

• Proofs

• Guided Proofs lead student through the initial steps

of constructing proofs and then utilizing the results.

• Writing Exercises

• Technology Exercises are indicated throughout

the text with .

• Exercises utilizing electronic data sets are indicated

by and found at www.cengagebrain.com.

Table of Contents Changes

Based on feedback from users, Section 3.4

(Introduction to Eigenvalues) in the previous

edition has been removed and its content has

been absorbed in Chapter 7 (Eigenvalues and

Eigenvectors).

Trusted Features

Section Objectives

A bulleted list of learning objectives provides you

the opportunity to preview what will be presented

in the upcoming section. For the Seventh Edition,

the section objectives are located by relevance at

the beginning of each section.

Theorems, Definitions, and Properties

Presented in clear and mathematically precise

language, all theorems, definitions, and properties

are highlighted for emphasis and easy reference.

x Preface

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Proofs in Outline Form

In addition to proofs in the exercises, some

proofs are presented in outline form, omitting

the need for burdensome calculations.

Discovery

Discovery helps you develop an intuitive

understanding of mathematical concepts and

relationships.

Technology Notes

Technology notes show how you can use graphing

utilities and software programs appropriately in the

problem-solving process. Many of the technology

notes reference the Online Technology Guide,

located at www.cengagebrain.com.

Chapter Projects

Two per chapter, these offer the opportunity

for group activities or more extensive homework

assignments, and are focused on theoretical

concepts or applications. Many encourage the

use of technology.

Preface xi

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Print

Instructor’s Solutions Manual

ISBN-13: 978-1-133-36538-9

The Instructor’s Solutions Manual provides worked-out solutions for all even-numbered

exercises in the text.

Media

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in the text. Solution Builder allows you to create customized, secure solutions printouts

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Diploma Computerized Testing

ISBN-13: 978-1-133-49045-6

Diploma is an easy-to-use assessment software containing hundreds of algorithmic

questions derived from the text exercises. With Diploma, you can quickly create,

customize, and deliver tests in both print and online formats.

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Instructor Resources

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